Arabian Journal for Science and Engineering (2019) 44:4151–4160https://doi.org/10.1007/s13369-018-3356-2

RESEARCH ART ICLE - C IV IL ENGINEER ING

Development of Improved Frequency Expressions for CompositeHorizontally Curved Bridges with High-Performance Steel Girders

Iman Mohseni1 · Yonghan Ahn2 · Junsuk Kang1

Received: 7 February 2018 / Accepted: 27 May 2018 / Published online: 5 June 2018© The Author(s) 2018

AbstractComposite curved I-girder bridges are often used in modern highway systems, but the open sections of I-girders meanthat these structures suffer from low torsional resistance. The curvature also results in quite complex behaviors due to thecoupled bending and torsional responses of curved I-girder bridges. High-performance steel, which adds strength, enhancesdurability and improves weldability, addresses both the economic and structural problems associated with curved bridges.However, as yet, there are no simplified design methods in the form of practical equations with which to optimize the designparameters of curved bridges and their dynamic behavior remains controversial. This study evaluated the effects of variousdesign parameters on the free vibration responses of curved HPS I-girder bridges. A sensitivity analysis of 278 prototypesimple-span and continuous bridges was conducted using CSIbridge software to create a set of simple, practical expressionsfor the fundamental frequencies of these structures.

Keywords Curved bridges · Frequency · Finite element method · High-performance steel

1 Introduction

Horizontally curved high-performance steel (HPS) I-girderbridges can offer an economical solution for modern high-ways where roadway alignments need a smooth and curvedtransition, reducing costs and controlling pollution due tocar emissions. In curved bridges, the centerlines of the girderwebs in the sections rising up from the abutments are notcollinear with the cords between the abutments. These eccen-tricities result in high torsional moments, causing high outof plane deformations and rotations in the bridge cross sec-tions [1–3]. High-performance steel (HPS) is often utilizedto address these problems as it provides more economic anddurable bridges, a significant advantage that enables engi-neers to design longer and shallower bridges. To enhancethe torsional resistance of open-section bridges, cross-framesand diaphragms are applied to interconnect girders in both

B Junsuk Kangwindkplus@hotmail.com

1 Department of Landscape Architecture and Rural SystemsEngineering, Seoul National University, Seoul 08826,Republic of Korea

2 School of Architecture and Architecture Engineering,Hanyang University, Ansan, Gyeonggi-do 15588, Republic ofKorea

curved and straight bridge superstructures [4–8]. Many stud-ies have sought to define the exact behavior of curved or bentbeams. In one of these earlier studies, Culver [9] developedseveral analytical models with which to investigate the inter-actions between curved girders and cross-frames. The effectof erection sequencing on the induced stress and deflectionshas also been studied in an attempt to enhance the designguidelines and improve the stability of the bridges [10–12].

To reduce the complexity of curved bridge design andimprove the capacity of design codes to predict the responsesof curved bridges, several researchers have tried to modelbridge superstructures as an assembly of simple systems,for example by applying the grillage technique, orthotropicplatemethods or other relatively simple forms [13–15]. How-ever, the accuracy of these methods remains problematicand the results produced may not be acceptable due to thesimplifications involved. Although finite element modeling(FEM) has been found to be a reliable method for evaluatingthe performance of curved bridges, the time and resourcesrequired mean that it is seldom feasible in a typical bridgedesign office, especially during the preliminary design stages[16,17].

Several analytical and numerical analyses have been per-formed to evaluate the dynamic behavior of compositeI-girder bridges. Christiano and Culver [18] developed a the-

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0

1

2

3

4

5

6

7

30 45 60 75 90

Freq

uency(Hz)

Span Length (m)

Wright and Green [29]Wood and Shepherd [31]Billing[30]Can�eni [32]Trilly [34]Dusseau [33]

Fig. 1 Fundamental frequencies obtained using various equations

oretical method using the differential equations governingthe free vibration of a simple-span curved girder, derivingtwo sets of equations to determine the fundamental frequen-cies of such bridges based on Lagrange multiplier concepts.Maneetes and Linzell [19] studied the effects of cross-frameand lateral bracing on the dynamic responses of a single-spancurved I-girder bridgeusingboth experimental andnumericalsolutions, while Yoon et al. [20] presented a finite elementformulation based on a free vibration analysis of compos-ite bridges, taking into account the stiffness as well as themass matrices of the curved beam elements. The findings ofextensive studies have indicated that grillagemethods are notin fact a reliable way to evaluate the dynamic behavior andfree vibration of bridges [3]. Barth and Wu [21] performeda simulation and an experimental study to capture the vibra-tion characteristics (bending frequencies and mode shapes)of straight HPS girder bridges.

Figure 1 shows the fundamental frequencies valuesdefined based on the equations proposed by variousresearchers for steel girder bridges. According to the graph,Trilly’s equation [22] generates the most conservative val-ues for the fundamental frequency, although all follow thesame trend with increasing span length. Some past reportson vibration analysis are quite controversial, as they suggestvery high values for the fundamental frequency. Wu [23]summarized these reports, concluding that fundamental fre-quencies are highly dependent on the geometry of bridgesand that beam theory and the simplified equations proposedso far are not capable of predicting the correct values. Nas-sif et al. [24] presented an investigation of vibration control(e.g., acceleration and velocity) in HPS bridges using anexperimentally validated three-dimensional dynamic com-puter model to evaluate both the deflection criterion anddepth-to-span (D:L) limitation produced by the currentdesign codes. They concluded that neither set of limita-tions could provide effective vibration control for steel girderbridges under normal truck traffic conditions and these lim-its therefore represent a poor method for controlling bridge

vibration or ensuring human comfort for those traveling overthem [23,25]. Several studies have shown that the currentAASHTO serviceability criteria may be insufficient for con-trolling bridge vibration and frequency-based limits aremorerational than span-based limits [25,26].

This paper presents the results of a simulation that appliesCSIbridge V20 [27] to design and analyze the fundamentalfrequencies of curved HPS I-girder bridges. A parametricstudy examines the variables that could influence the freevibration characteristics of typical composite HPS I-girderbridges. The results from a comparison study indicated thatexisting frequency prediction equations are not sufficientlyaccurate for curved composite I-girder bridges. Hence, non-linear regression methods are used to derive two sets of morereliable and practical expressions with which to estimate thefundamental frequencies of curved HPS I-girder bridges thatcapture the effects of all the important parameters.

2 Finite Element Modeling and Verification

In this section, the general attributes of the proposed finiteelementmodeling (FEM)usingCSIbridge [27] are described.The results obtained from two field tests and a laboratorystudy are used to verify the effectiveness of the proposedmodeling technique for curved HPS bridges.

2.1 Bridge Sections

Afour-node shell elementwith six degrees of freedomat eachnode that combines membranes and plate-bending behaviorwas used to model the bridge’s concrete slab and steel gird-ers. Each element has its own local coordinate system fordefining material properties and loads, and for interpretingthe output. Stresses and internal forces and moments in theelement local coordinate systemwere evaluated at the 2-by-2Gauss integration points and extrapolated to the joints of theelement. The bridge modeler in CSIbridge [27] was used tomodel the prototypebridge.This option takes into account thefull composite action between the reinforced concrete slaband the HPS I-girders at the serviceability limit state, as rec-ommended by AASHTOLRFD [6]. Cross section propertieswere defined for all frame-elements and diaphragm proper-ties imported (X or V types) and manually defined (built-upplate sections). The parapets were modeled as a single frameelement. Note that it is critical to place the connecting nodesof the parapets at the centroid of the barrier section.Addition-ally, the parapets were attached to the deck using a series ofrigid links in order to contribute to the stiffness of superstruc-ture. Figure 2 shows the parapet model assumptions applied.

The discretization of the models was performed usingthe auto-meshing option, defining a good expectation ratiofor all elements. The boundary conditions for simple-span

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Fig. 2 Parapet modeling assumptions

Fig. 3 Selected boundary conditions

and continuous curved bridges were defined based on theproposed tangential bearing arrangement recommended bySamaan et al. [28], shown in Fig. 3. In the elastic range ofstructural behavior, any slab cracking present exerts only anegligible effect on the bridge responses [29]; hence, the slabwas assumed to be uncracked and steel reinforcements weretherefore not modeled.

2.2 Verification of Finite Element Models

2.2.1 Colquitz River Bridge

The results of dynamic field tests on the Colquitz river bridgein British Columbia, Canada [30] were selected to verifythe proposed FEM technique. The bridge has five spans(2 × 14m+ 2 × 17m+ 18m), each composed of six steelgirders W33 × 141, and a bridge width and slab thicknessof 11.9 and 0.10m, respectively. The diaphragms and cross-braces members are made ofU-shaped steel (MC18× 42.7)and spaced 3.5 and 4.5m apart along the end and interme-diate spans, respectively, of the bridge. For the ambient test,six force-balanced accelerometers were used to collect andprocess acceleration data during the day and under varyingtraffic conditions. Pullback tests were carried out by load-ing the bridge bents with a special assemblage using a force

of 90kN, and then quickly releasing the applied load toinduce free vibrations in both the longitudinal and transversedirections. Nearby abandoned railway bridge piers providedadequate anchor points for the pullback testing. The softwarecapabilities of hybrid bridge evaluation system (HBES) wereused to analyze records on site shortly after the data werecollected. The resulting torsional and vertical modes exhib-ited some coupling, which is typical in this type of analysiswhen periods are close in value [30]. In order to comprehen-sively verify the results of the FEM, the entire bridge wassimulated in the present study using the CSIbridge packagesoftware according to the prescribed modeling technique.The first three vertical frequencies obtained by the bridgemodel (with parapet effects) were 6.12, 7.48 and 8.75Hz,differing from the field test frequencies of 5.95, 7.14 and8.35Hz by about 3.50, 4.54 and 4.72%, respectively. With-out modeling the parapet, the first three vertical frequenciesobtained by the FEM method were 6.03, 7.25 and 8.50Hz,which indicates that including the parapet shifts the resultsby 1.32, 1.52 and 1.76%, respectively. These results indicatethat the FEM technique adequately predicts the vibrationresponses of composite girder bridges. Since the funda-mental frequency (the first natural frequency) is the mostpronounced, the parapets may not noticeably influence thevertical frequency of these bridges.

2.2.2 Laboratory Test of a Quarter-Scale Model Bridge

Additional verification was performed through experimentaltests of a quarter-scale model of a 4.5m long simple-spanstraight I-beam bridge with five scaled girders at a spacingof 0.45m and width of 1.80m [31]. These I-beams had thefollowing dimensions: flange thickness of 4.80mm, flangewidth of 58mm, web thickness of 3.5mm and total height of203mm. Channels with dimensions of 25× 9.5× 0.30mmand a spacing of 15.8cm were used as shear connectors.Here, solid diaphragms were applied at the abutments andspaced at one-third intervals along each span, consisting of76 × 50 × 52mm angles welded to the beam webs about13mm below the top of the beam. A comparison of the FEMand experimental results for the mid-span deflections of allgirders under four 20kN concentrated loads at the mid-spanand symmetric about the longitudinal axis of bridge is shownin Fig. 4.

Although the experimental and FEM results comparefairly well, the FEM results are slightly higher than thoseobtained in the laboratory, varying by up to 1.5, 4.6 and 3.4%for the external, intermediate and central girders, respec-tively. This discrepancy is likely because the experimentalloads were not localized at an exact position but rather dis-tributed over the entire steel supporting plate. The samephenomenon has also been reported by Wegmuller [32] ina comparison between FEM and experimentally obtained

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Fig. 4 Deflection distribution in the bridge girders at themid-span crosssection

results. This suggests that the proposed FEM techniqueaccurately predicts the response of I-girder bridges to staticloading.

2.3 Geometric andMaterial Properties of Bridges

High-performance steel (HPS) bridges need less steel dueto their higher yield stress, but consequently the live-loaddeflections of these bridges are more likely to exceedAASHTO’s deflection limits [5]. As design optimizationmust take into account the AASHTO limits on performanceand economy, an important first step is to select a set of keyparameters and establish a matrix that covers a wide range ofsteel bridge types. Based on this database, bridges can thenbe designed and optimized for various combinations of theseparameters to capture a least weight approach using CSIb-ridge [27]. The initial designs are performed by neglectingthe AASHTO criteria for deflection in order to design girdersthat meet other AASHTO LRFD strength and serviceabil-ity criteria for curved bridges. Any girders that then fail theAASHTO live-load deflection criteria must be redesigned.Table 1 shows the geometric properties and material param-eters for three prototype bridges. Figure 5 shows a typical30m span length bridge modeled in CSIbridge [27].

Four different span lengths (L), 30, 45, 60, and 90m, wereconsidered in this investigation, with span-to-radius (k) val-

Fig. 5 FE model of a 30m span length bridge with k = 0.1

ues for the slenderness ratio increasing from 0 to 1.20 at 0.2intervals. Three types of cross section, with girder spacingsof 2.8, 3.5, and 2.5m, were utilized, along with the Ameri-can Iron and Steel Institute (AISI) standard parapet design,which has a height of 0.87m and bottom and topwidths of 0.4and 0.15m, respectively. The prototype bridges were mod-eled using three different high-performance steel materials:HPS 50W (Fy = 345Mpa), HPS 70W (Fy = 485Mpa) andHPS 100W (Fy = 620Mpa). The concrete deck was mod-eled with a modulus of elasticity of 28,000Mpa, a Poisson’sratio of 0.2 and a density of 24kN/m3. The concrete deckthickness was 0.20m for all the cross sections.

3 Sensitivity Analysis

A sensitivity analysis provides useful information regard-ing the influence of various parameters on the free vibrationresponses of bridges. The goal here is to characterize thekey parameters affecting a bridge’s natural frequency so thatthey can be taken into account in the development of effectiveexpressions for natural frequencies. The investigation vari-ables are the span length, the span-to-radius curvature ratio(k), the number of girders, the girder spacing and the numberand arrangement of the cross-braces. A detailed discussionof each of these parameters is presented in this section, andthe key parameters are identified.

Table 1 Characteristics of threeprototype curved I-girderbridges

Set L (m) HPS types (W) L/D ratio S (m) No. girder W (m)

1 30, 45, 60, 75, 90 50 20 2.80 5 13.0

2 30, 45, 60, 75, 90 70 25 3.50 4 13.0

3 30, 45, 60, 75, 90 100 30 2.50 4 9.50

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0

2

4

6

8

10

12

14

2 3 4 5 6 7 8 9

Freq

uency(Hz)

No. cross braces

f1 f2 f3f4 f5

Davidson’s equa�on

Fig. 6 Effect of the number of cross-bracing lines on a bridge’s naturalfrequencies

3.1 Effect of the Number of Cross-braces

Curved girder bridges have a high torsional moment dueto their curvature. The open nature of these bridges resultsin low torsional resistance to cracking, so cross-braces areapplied to enhance their torsional stiffness. It is thus vital todetermine the effect of this variable on the bridge’s free vibra-tion responses. The effect of the number of cross-bracinglines on the first natural frequency of a 30m long bridge witha curvature ratio of 1.20 is shown in Fig. 6. The minimumrequired number of cross-braces obtained from Davidson’sequation [33] is also indicated. Although the number ofbraces appears to have only an insignificant effect on thefirst and second frequencies and the vertical mode shapes ofthe bridge, it does decrease the natural frequency values forthe higher modes of vibration. Thus, for all the prototypebridges modeled in this study, cross-braces are placed 6mapart, with a minimum of five bracing lines for each span.

3.2 Effect of the Cross-brace Type

In order to study the effect of the various types and stiffnessesof bracing systems on the free vibration responses of curvedHPS bridges, several different bracing configurations weremodeled for a 90m long bridge with a curvature ratio of0.60. The following bracing systems were considered: X-type, V-type, inverted V-type, single beam and solid plates;the results are shown in Fig. 7. Once again, although thetype of bracing systems had an insignificant impact on thefirst natural frequency of the bridge, it did have an importantimpact on the higher frequencies. For instance, the secondnatural frequency of a bridge with X-type bracing was up to12, 48 and 54% higher than bridges with V-, inverted V- andsingle beam bracing systems, respectively. The effects of thestiffness of the cross-bracing and the end diaphragms on thefree vibration of the bridge are also shown in Fig. 7.

These results suggest that the stiffness of secondary mem-bers has a significant influence on the bridge’s fundamental

Fig. 7 Effect of cross-brace type on a bridge’s natural frequencies

frequencies. For example, enhancing the thickness of thesolid steel plates from 10 to 25mm changes the first andsecond frequencies by up to 2%. However, although increas-ing the plate thickness also increases the natural frequenciesof the higher modes, it should be avoided due to possiblebrittle connection failures of the diaphragms under seismicloads. Thus, it is preferable to adjust the thickness of thesteel X-bracing rather than the plate thickness to enhancethe torsional resistance of both single-span and continuousbridges.

3.3 Effect of the Span Length

Many researchers have described the importance of spanlength for the dynamic responses of straight bridges. Forexample, the current North American codes [5,6] offer equa-tions for the dynamic impact factor of bridges as a function ofspan length. Wright and Walker [34] developed the follow-ing theoretical equation based on beam theory to determinethe fundamental frequencies. Note that this equation incor-porates the span length as an important factor:

fsb = π2L2

√Eb · Ib · g

w(1)

where Eb · Ib and w are the flexural rigidity and the weightper unit length of the composite steel girder, respectively.Figure 8 shows the effect of span length on the natural fre-quencies of bridges with a slenderness ratio (L/D) of 25 anda curvature ratio of 0.20. Increasing the span length from 30to 90m decreases the natural frequencies by up to 65%. Froma practical standpoint, the span length is thus an importantparameter affecting the dynamic response of a curved bridge.

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0

2

4

6

8

10

30 45 60 75 90

Freq

uency(Hz)

Span Length (m)

k=0.2, L/D=25 f1f2f3f4

Fig. 8 Effect of span length on the fundamental frequencies of curvedbridges (k = 0.20, L/D = 25)

0

1

2

3

4

5

6

7

8

9

3 4 5 6 7

Freq

uency(Hz)

Number of Girders

L=60m, k=0.60f1f2f3f4f5

Fig. 9 Effect of the number of girders on a bridge’s natural frequencies

3.4 Effect of the Number of Girders

Analyzing the performance of a 60m long curved bridgewitha curvature ratio of 0.6 to investigate the effect of the numberof girders on the bridge’s natural frequencies revealed thatalthough increasing the number of girders results producedhigher torsional stiffness in the structure, the natural fre-quencies of the bridge’s superstructure were enhanced onlyslightly. The results, presented in Fig. 9, show that the firstand second frequencies increase by approximately 3% as thenumber of girders increases from three to seven, for example.

In practice, this does not influence the free vibrationsufficiently to justify its inclusion as an important modelparameter. These relatively small changes are due to the factthat the prototype bridges have almost the same flexural stiff-ness, irrespective of the number of girders. Previous researchhas also indicated that the number of girders has no impacton the natural frequencies of straight bridges [35].

3.5 Effect of the Span-to-Radius Curvature Ratio

Figure 10 shows the relationship between the first naturalfrequencies (f1) and curvature ratios (k = L/R) of fivebridges with lengths of 30, 60 and 90m and various slen-

0.5

1

1.5

2

2.5

3

3.5

4

0.2 0.4 0.6 0.8 1 1.2

FirstF

requ

ency

(Hz)

k=L/R

L=30m, L/D=25, 50WL=30m, L/D=25, 70WL=30m, L/D=30, 70WL=60m, L/D=20, 70WL=60m, L/D=20, 50W

Fig. 10 Effects of the bridge curvature ratio on the fundamental fre-quency

derness ratios (L/D). The graph indicates that the curvatureratio has an important effect on the free vibration of curvedsuperstructures, with an increase in the curvature ratio sig-nificantly reducing the natural frequencies. For instance, forthe 60 and 90m bridges, the natural frequencies decreaseby up to 56 and 39%, respectively, when the span-to radiusratio increases from 0 to 1.20. The curvature ratio also has asignificant impact on the mode shapes.

Figure 11 shows the first three mode shapes for bothstraight and curved bridges with 30m long spans. Thesereveal that any curvature in a bridge exerts a higher torsionalimpact on mode shapes. For example, the first mode shapefor a straight bridge is purely flexural, but in a curved bridge itconsists of a combination of flexural and symmetric torsionalmodes. Thus, the effect of the curvature ratio on both themode shapes and frequencies will be significant and shouldthus be included in the parametric study.

3.6 Effect of Slenderness (Span-to-Depth) Ratio

AASHTO (5) specifications limit the value of the slendernessratio to a maximum of 25 in order to control the maximumlive-load deflection through controlling the bridge’s bendingstiffness. However, this approach became somewhat con-troversial with the introduction of high-performance steel(HPS),which has approximately 40%higher yield stress thanconventional steel and hence allows engineers to design shal-lower and longer spans. Since HPS bridges are more likelyto suffer from large deflections, the slenderness ratio playsa significant role in the dynamic behavior of these systems.The effects of different slenderness ratios on the natural fre-quencies of a 30m long bridge are presented in Fig. 12. Thenatural frequencies decrease almost linearly as the L/D ratioincreases, irrespective of the curvature ratio. For example,in a bridge with a curvature ratio of 0.80, the first naturalfrequency decreases by approximately 28% when the L/Dratio increases from 20 to 30. From a practical standpoint,the impact of the slenderness ratio is considerable and should

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Fig. 11 First three modes ofvibration for straight and curvedbridges. a Straight bridges and bcurved bridges

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

035202

Freq

uency(Hz)

Slenderness ra�o (L/D)

simple supported bridge, L=30m k=0.20k=0.40k=0.80k=1.20

Fig. 12 Effect of slenderness ratio (L/D) on a bridge’s natural fre-quency

thus be taken into account in any parametric study of curvedHPS I-girder bridges.

4 Empirical Expressions for FundamentalFrequency

As the comparison of the various previous studies that haveinvestigated this issue presented in Fig. 1 shows, none of the

proposed equations adequately predict the fundamental fre-quency of curved composite girder bridges constructed usinghigh-performance steel. While Wood and Shepherd’s equa-tion [36] estimates highly conservative values, the modelsproposed by Dusseau [37] and Wright and Walker [38] bothsignificantly underestimate the fundamental frequencies ofsuch bridges. To address this problem, the data for 180 sim-ple supported and 85 two- and three-span continuous bridgeswere collected and analyzed to develop a set of proposedexpressions to describe their fundamental frequencies. Theproposed expressions take the following form:

fc·b = ψ · fsb (2)

where fsb is the fundamental frequency of a straight bridgefrom Eq. (1). A modification factor, ψ , is applied to takeinto account the effect of the parameters from the previoussection that were found to affect the free vibration of curvedHPS I-girder bridges. A regression analysis using a statisti-cal computer package for best fit based on the least squaresmethod for nonlinear data was used to deduce appropriatemodification factors as follows:

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Fig. 13 Comparison of theresults obtained by the proposedexpression and the FEManalysis. a Simple-span and bcontinuous span

(a) (b)

R² = 0.9739

0

1

2

3

4

0 1 2 3 4

Freq

uencyEq.(3

)Frequency FEM Analysis (Hz)

R² = 0.9772

0

1

2

3

4

0 1 2 3 4

Freq

uencyEq

.(4)

Frequency FEM Analysis (Hz)

Table 2 Comparison betweenthe FEM results and thoseobtained using the proposedexpressions

No. spans L/D L (m) L/R fsb (Hz) ψ Frequency (Hz)fc·b = ψ · fsb FE

1 20 30 1.0 0.5966 2.240 1.337 1.375

1 25 30 0.80 0.6865 1.930 1.328 1.281

1 30 30 1.0 0.5910 1.980 1.172 1.102

2 30 30 1.20 0.8435 2.290 1.936 2.020

2 30 60 0.20 1.2063 1.050 1.275 1.250

• Simple-span bridges

ψ = 1.160×e−0.82k×e0.018S×(0.80+ 0.008 L

D

)(3)

• Continuous bridges

ψ = 1.137× e−0.37k × e0.005 LD (4)

It should be recognized that the effect of the slendernessratio in the proposed expressions for the modificationfactor is negligible because the impact of the span lengthand depth has already been taken into account in Eq. (1).The results of the verification analyses for the proposedexpressions are shown in Fig. 13 and Table 2.

Examining the first natural frequency for both simple-spanand continuous curved bridges reveals that the differencebetween the FEM analyses results and those obtained usingthe expressions proposed here are within 1%. The high coef-ficient of determination, R2, obtained in both cases indicatesthe minimal variation in the results calculated by the pro-posed expressions and the FEM analysis.

5 Conclusions

This study has presented a detailed numerical analysis ofthe free vibration characteristics of horizontally curved HPS

I-girder bridges. This numerical approach includes an exten-sive sensitivity study of simple supported and continuousbridges to determine the effect of a number of different vari-ables on the natural frequencies and mode shapes of thebridge structures. Empirical expressions for the fundamentalfrequencies of such bridges were derived that are suitable foruse in design codes and engineering offices. The proposedexpressions are a function of the natural frequency obtainedfromflexural beam theory and are hence applicable to bridgeswith various slenderness ratios. Based on the results obtainedfrom the sensitivity study, the following conclusion can bederived:

• The slenderness ratio significantly affects the naturalfrequency of such bridges: the fundamental frequencydecreases with increasing span-to-depth ratio.

• The magnitude of the fundamental frequency decreaseswith increasing span-to-radius curvature ratio for hor-izontally curved I-girder bridges. The curvature ratioalso has a significant effect on the bridge’s vibrationmodes due to the high contribution of the torsional effectsinduced in curved superstructures.

• Cross-braces between the abutments increase the tor-sional stiffness of open-section bridges. It is recom-mended that the maximum spacing for bracing lines belimited to 6m in order to exert a meaningful effect onbridge responses.

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Acknowledgements The work reported herein was supported by aGrant (18CTAP-C132633-02) funded by the Ministry of Land, Infras-tructure and Transport (MOLIT) of the Korean Agency for Infras-tructure Technology Advancement (KAIA), the National ResearchFoundation of Korea (NRF) Grant (NRF-2016R1C1B1015711) fundedby theKorea government (Ministry of Science, ICT&Planning) and the2017 Seoul National University Invitation Program for DistinguishedScholars. This financial support is gratefully acknowledged.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

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Development of Improved Frequency Expressions for Composite Horizontally Curved Bridges with High-Performance Steel GirdersAbstract1 Introduction2 Finite Element Modeling and Verification2.1 Bridge Sections2.2 Verification of Finite Element Models2.2.1 Colquitz River Bridge2.2.2 Laboratory Test of a Quarter-Scale Model Bridge

2.3 Geometric and Material Properties of Bridges

3 Sensitivity Analysis3.1 Effect of the Number of Cross-braces3.2 Effect of the Cross-brace Type3.3 Effect of the Span Length3.4 Effect of the Number of Girders3.5 Effect of the Span-to-Radius Curvature Ratio3.6 Effect of Slenderness (Span-to-Depth) Ratio

4 Empirical Expressions for Fundamental Frequency5 ConclusionsAcknowledgementsReferences